Question

A nanoparticle containing 6 atoms can be modeled approximately as
an Einstein solid of 18 independent oscillators. The evenly spaced
energy levels of each oscillator are 4e-21 J apart. Use k = 1.4e-23
J/K.

(a) When the nanoparticle's energy is in the range 5(4e-21) J to
6(4e-21) J, what is the approximate temperature? (In order to keep
precision for calculating the heat capacity, give the result to the
nearest tenth of a degree.)
K
(b) When the nanoparticle's energy is in the range 8(4e-21) J to
9(4e-21) J, what is the approximate temperature? (In order to keep
precision for calculating the heat capacity, give the result to the
nearest tenth of a degree.)
K
(c) When the nanoparticle's energy is in the range 5(4e-21) J to
9(4e-21) J, what is the approximate heat capacity per atom?
J/K
Note that between parts (a) and (b) the average energy increased
from "5.5 quanta" to "8.5 quanta". As a check, compare your result
with the high temperature limit of 3k, where k = 1.4e-23 J/K.